Exercise 3, § 17

Let $E$ be a field, $G$ a group of automorphisms of $E$ and $K$ the field of invariants of $G$. a) Show that if there exists no normal subgroup of finite index in $G$ other than $G$, then $E$ is a regular extension of $K$. b) Deduce from a) that if $G$ is isomorphic to a finite product of infinite simple groups, then $E$ is a regular extension of $K$.